How can I parent a mesh to a vertex group?
Add a Surface Deform modifier to the eye lashes object. Then set the modifier target to the head. Click the 'Bind' button.

Now any shapekeys or armatures that affect verts in the head mesh, will also affect the eyebrow mesh.



If you make any edits to the amount of vertices in either mesh, you need to re-bind in the modifier. 

It should be noted that the modifier doesn't use vertex groups like you asked for specifically. Instead, it maps the vertices of its parent mesh, to the closes points of the target mesh. Those points may not exactly align with vertices.
However, in your specific use case it seems like the eyelash mesh, and the eye lid mesh match almost exactly. So, this shouldn't cause any noticeable render issues.